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arXiv · 2308.09674

Microscopic derivation of a Schr\"odinger equation in dimension one with a nonlinear point interaction

Riccardo Adami·Jinyeop Lee
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Abstract

We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive three-body potential. Assuming a simultaneous mean-field and short-range scaling with the short-range proceeding slower than the mean-field, and choosing an initial fully condensed state, we prove propagation of chaos and obtain an effective one-particle Schr\"odinger equation with a defocusing nonlinearity concentrated at a point. More precisely, we prove the convergence of one-particle density operators in the trace-class topology and estimate the fluctuations as superexponential. This is the first derivation of the so-called nonlinear delta model, widely investigated in the last decades, as a phenomenological model for several physical phenomena.

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Riccardo Adami, Jinyeop Lee. 2023-08-18. Microscopic derivation of a Schr\"odinger equation in dimension one with a nonlinear point interaction. https://doi.org/10.1016/j.jfa.2025.110866

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